Answer
$\frac{5}{11}$
Work Step by Step
$0.\overline{45}$ can be represented by the sum of the infinite geometric series with first term $a_1=0.45$ and common ratio $r=0.01$. An infinite geometric series converges if and only if $|r|\lt1$, where $r$ is the common ratio. If it converges, then it equals $\frac{a_1}{1-r}$ where $a_1$ is the first term.
Here $0.01\lt1$.
Hence the sum: $0.\overline{45}=\dfrac{0.45}{1-0.01}=\frac{5}{11}$
